Convergence Speed of Binary Interval Consensus

Moez DraiefImperial College London

Milan VojnovićMicrosoft Research

IEEE Infocom 2010, San Diego, CA, March 2010

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Binary Consensus Problem0

10

1

0

0

1

1

0

1

0

• Each node wants answer to: was 0 or 1 initial majority?

0

• Requirements:local interactionslimited communicationlimited memory per node

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Related Work• Hypothesis testing with finite memory

(ex. Hellman & Cover 1970’s ...) – But typically not for dependent observations in network settings

• Ternary protocol (Perron, Vasudevan & V. 2009)– Diminishing probability of error for some graphs– Ex. complete graphs – exponentially diminishing probability of error with

the network size n; logarithmic convergence time in n

• Interval consensus (Bénézit, Thiran & Vetterli, 2009)– Convergence with probability 1 for arbitrary connected graphs– Limited results on convergence time

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Our Problem

Q: What is the expected convergence time for binary interval consensus over arbitrary connected graphs?

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Binary Interval Consensus• Four states

0 1e0 e1

e00

e0 0

e10

e0 0

0 1

e0e1

e0 e1

e0e1

e0

e11

1 e1 1

e11

• Update rules– Swaps– Annihilation

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Outlook

• Upper bound on expected convergence time for arbitrary connected graphs

• Application to particular graphs– Complete– Star-shaped– Erdös-Rényi

• Conclusion

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General Bound on Expected Convergence Time

• Let for every nonempty set of nodes S, :

• Each edge (i, j) activated at instances a Poisson process (qi,j)

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General Bound on Expected Convergence Time (cont’d)

• Without loss of generality we assume that initial majority are state 0 nodes• a n = initial fraction of nodes in state 0, other nodes in state 1, a > 1/2

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General Bound on Expected Convergence Time (cont’d)

• Key observation: two phases– In phase 1 nodes in state 1 are depleted– In phase 2 nodes in state e1 are depleted

• Phase 1

1 if node i in state 1 1 if node i in state 0

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Phase 1

• Dynamics:

Sk = set of nodes in state 0

• The result follows by using a “spectral bound” on the expected number of nodes in state 1

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Outlook

• Upper bound on expected convergence time for arbitrary connected graphs

• Application to particular graphs– Complete– Star-shaped– Erdös-Rényi

• Conclusion

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Complete graph• Each edge activated with rate 1/(n-1)

• Inversely proportional to the voting margin• Can be made arbitrary large!

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Complete graph (cont’d)• The general bound is tight

• 0 and 1 state nodes annihilate after a random time that has exponential distribution with parameter cut(S0(t), S1(t)) / (n-1)

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Star-shaped graph• Each edge activated with rate 1/(n-1)

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Star-shaped graph (cont’)

• By first step analysis:

• Same scaling, different constant

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Erdös-Rényi graph• Each edge age e activated with rate Xe /npn

where Xe ~ Ber(pn)

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Erdös-Rényi graph (cont’d)

• For sufficiently large expected degree, the bound is approximately as for the complete graph– In conformance with intuition

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Conclusion• Established a bound on the expected convergence time of

binary interval consensus for arbitrary connected graphs

• The bound is inversely proportional to the smallest absolute eigenvalue of some matrices derived from the contact rate matrix

• The bound is tight– Achieved for complete graphs– Exact scaling order for star-shaped and Erdös-Rényi graphs

• Future work– Expected convergence time for m-ary interval consensus?– Lower bounds on the expected convergence time?